Translation is a transformation in which every point of an object moves the same distance in the same direction. If a translation is on a coordinate grid, the moves parallel to the x-axis and parallel to the y-axis must be specified.

A translation has these properties:

The object and image are congruent.

The orientations of the original and the image are the same.

The inverse of any translation is an equal move in the opposite direction.

Example:

A translation sends ABCD to

EFGH

PQRS

TUVW

Describe each translation in detail.

The translation is 4 units left

The translation is 4 units down

The translation is 4 units left and 4 units down.

What this might look like in the classroom

Question 1:

The following diagram shows a blue triangle and its (white) translation.
Describe the translation in detail.

Solution 1:

The white triangle is a translation 4 squares right and 1 square up

Note that a common misconception would be that the shape had moved one square right – but this is the number of squares between the two shapes. Learners should use tracing paper (or dynamic geometry software) to actually carry out the translation and see that each vertex (and therefore the shape) has moved four squares right.

Question 2:

Look at rectangle A here:

Translate rectangle A 2 squares right and 4 squares down. Label the new rectangle ‘B’. Now translate B 3 squares left and 1 square up. Label this rectangle ‘C’. What single translation would move rectangle A to rectangle C?

Solution 2:

1 square left and 3 squares down

Taking this mathematics further

Translation is one of the four basic transformations, the others being reflection, rotation and enlargement. Translations, reflections and rotations result in congruent shapes; i.e. they have the same shape and size, but the position varies. Enlargement of a shape results in a similar shapes; mathematically similar shapes have the same shape, but a different size.

A translation can be described in terms of a vector. For example, the translation 4 squares right and 1 square up could be described by the vector ( 4 1 ) ,

while the translation 1 square left and 3 squares down would be
(
-1
-3
)
.

Note that the top number represents the horizontal movement. The vector can be drawn by joining any pair of corresponding vertices on the shape and its translation, with an arrow indicating the direction from the original shape (object) to its translation (image), as shown below:

Making connections

Encourage learners to look for examples of translation outside of the classroom. There are plenty of opportunities for them to notice them. Particular examples of note are:

Wallpaper patterns; of which there are seventeen different groups relying on basic transformations – including translation.

Frieze patterns; which are defined as 2D designs that are repetitive in one direction only. There are seven different groups, again relying on basic transformations, and car tyre treads almost always correspond to one of these.

Rangoli patterns: traditional Indian geometrical patterns based on reflection, rotation and translation. Often displayed during the Hindu festival of lights: Diwali.